How to make overlayed composited images stay intact while changing render resolution?
Scale node set to Scene Size mode will scale image accordingly to the render resolution:


  Scene Size
      Sizes an image to the size of the final render resolution for the scene. For example, rendering a scene at the standard 1080p resolution but setting the render percentage at 50%, will produce a 1080p image with the scene scaled down 50% and leaving the rest of the image as alpha.


If feeding it with original image and then plugging the output of the node into nodes shown in the setup above it will work for different render resolutions:



The node setup will be the same with the only difference that now it has 1 additional Scale node.  Fit option used here which is the easiest one for preserving image original ratio: